Question

1) Carlos has borrowed $8,000 for 8 years at 6% compounded semi-annually. He will repay interest every 6 months plus principal at maturity. He will also deposit X every 6 months into a sinking fund paying 5% compounded semi-annually to pay off the principal at maturity. a) Find X. Carlos goes bankrupt at the end of year 6, just after making his interest payment and sinking fund deposit. The bank confiscates the money in the sinking fund but gets no further payments. b) How much money does the bank lose as a result of the loan default at the end of year 6? c) Over the lifetime of the loan, how much money did the bank collect? Was it more or less than the amount of the original loan? d) Assuming the bank re-invested all of Carlos payments at 6% (compounded semi- annually), how much money does the bank have at the end of 6 years? What is the equivalent yield (compounded semi-annually) they made on their initial loan?

Answer 1

a) This can be done using below formula which can be derived as we do derivation of annuity formula.

P = Ai/((1+i)" – 1)

A, accumulated amount = 8000

i, interest rate for the period = 5%/2 = 2.5% because interest rate calculated at semi annual rate

n, period = 8x2 = 16 periods

Therefore P = (8000x 2.5%)/((1+0.025)¹⁶ -1) = 412.79

Hence X = 412.79

b) Amount in Sunk fund at the end of 6th Year is

Using the same formula,

P = Ai/((1+i)" – 1)

we can rearrange to

A = P((1+i)" - 1)/i

Now, we have to find A

P = X = 412.79

i = 2.5%

n = 6X2 = 12

Putting values in formula

A = 412.79(1.025¹² -1)/.025 = $5694.67

Therefore loss to the bank = 8000- 5694.67 = $2305.33

c) Interest paid by Carlos after every 6 months = i X Principal = 6%/2 X 8000 = 3% of 8000 = $240

Therefore Total Money got from Interest = 240x12 = 2880

total money bank collected = Interest + Sunk fund = 2880 + 5694.67 = 8574.67

This is more than $8000 by $574.67

d) Interest after 6 years =

P((1+i)" - 1)

i = 6%/2 =  3%

n = 6x2 = 12

Therefore Interest = 8000(1.03¹² -1) = $3406.087

Principal received = $5694.67

Total Money bank has = 9100.757

To calculate semi annual yield

A= P(1+i)"

9100.757 = 8000(1+i_annual/2)¹²

^{=> i_{annual = 2X ((9100.76/8000)}1/12} - 1 ) = 2.16%